Ultrasonic inspections of reactor components usually require substantial amounts of power from the transducers producing the signals. This power is presently achieved by driving the ultrasonic transducer with electronic power amplifiers that are expensive, bulky and unreliable to some degree. In addition, the transducer signal typically has a sizeable bandwidth, which distributes the sonic energy over a considerable range of frequencies.
There are substantial benefits that accrue to narrow-band ultrasonic sources of short duration in time (so-called "tone-burst" signals). This can also be achieved to some degree using electronics components and conventional sources known to the art, but such a process is expensive and less than ideal technically. Rather, some inexpensive, reliable and compact means of generating the ultrasonic energy with narrow bandwidth would simplify the design and operation of many inspection systems.
Analysis of the interaction of a viscous and compressible fluid with an infinitely long, rigid cylindrical shell, whose variable radius fluctuates sinusoidally about a mean value along its axis, has lead to the discovery of a solution to the problem of amplifying ultrasound over long paths inside vessels of nuclear reactors. That analysis is set forth briefly below.
Ultrasonic waves propagating in fluid-filled ducts are essentially compressive in nature, in the sense that the fluid motion is parallel to the direction of propagation. The three-dimensional propagation of sound in fluid-filled ducts has been studied in detail theoretically and is now understood for thin, flexible duct walls and viscous fluids. It is clear that one possible mode of propagation is axisymmetric with negligible damping (at sufficiently high temperature), wherein the radial mode shape is inconsequential. These findings have led to the disclosure of devices useful as ultrasonic waveguides at frequencies of a few megahertz.
As a wave propagates axially in viscous fluid in a rigid cylindrical shell having a wavy wall, internal energy of the fluid is continuously and reversibly exchanged with the intensity of the wave. Ideally there is no net increase of entropy in the process, since heat transfer is neglected between the fluid and the wall of the rigid cylindrical shell. Under ordinary conditions, in which the system parameters take on arbitrary values, the wavy wall of the shell has no unusual effect on the fluid motion, and the damped wave propagation is readily described by the well-known fluid momentum and continuity equations for adiabatic, isentropic flow.
However, there exists a special case for which the wave intensity grows with distance along the duct, due to interaction of the fluid with the periodic wall. This occurs when the wavelength of the wall perturbation is about half that of the fluid wave. The transformed equation describing the wave motion is the Mathieu-Hill equation, whose solutions (Floquet functions) are known to possess regimes of instability for certain ranges of the parameters in the equation. The invention lies in the recognition that a fluid instability, akin to parametric amplification, can result from the interaction of the fluid with the periodic wall of the shell.
From a lengthy analysis, it can be shown that the wave velocity scalar potential .phi. is a solution of the Mathieu-Hill equation in the form: ##EQU1## where q=.epsilon./R&lt;&lt;1; R is the shell nominal (or mean) radius; .epsilon. is the amplitude of the periodic wall perturbation (variation in radius); .gamma. is the ratio of the wall perturbation wavelength to the sonic wavelength; y=(1-q)kz is the normalized axial coordinate; k is the ultrasonic wavenumber; z (&gt;0) is the axial coordinate; and .alpha. is a constant (.about.1).
This deceptively simple equation results when the viscous damping is negligible and the bulk fluid temperature is nearly constant (isothermal fluid for all z). In particular, for .alpha.=1 and .gamma.=2, series solutions are known of the general form: EQU .phi..sub..nu. (y)=e.sup.i.nu. y P(y); .nu.=.nu.(.alpha.,q)
and the solutions display the property: EQU .phi..sub..nu.(y+n.pi.)=e.sup.i.nu.(y+n.pi.) P(Y)
Thus, there is periodicity of .pi.(the Floquet condition), consistent with the nature of the periodic coefficient in the differential equation above. The series representations of the Floquet solutions are known as Mathieu functions.
The characteristic exponent .nu. is generally complex and depends on the parameters .alpha. and q in a complicated way. Numerical means exist for computing .nu.(.alpha., q), including expansions for fixed .nu., q when q&lt;&lt;1 and .nu. is non-integral (the case here); e.g., ##EQU2##
Neglecting terms of order greater than q.sup.2, and taking .alpha.=1, the characteristic exponent is a root of ##EQU3##
This quartic equation has two complex roots whose real parts are positive: ##EQU4##
The solution with positive real part and negative imaginary part represents an exponentially increasing solution as the wave propagates toward +z. We say that the solution is unbounded for this particular choice of the parameters; that is, parametric amplification results. On the other hand, the solutions with negative real part and positive imaginary part represent waves traveling toward -z and also growing in amplitude. A typical solution for a growing wave is illustrated in FIG. 1.
Clearly, this amplification only results for exceptional combinations of the parameters in the Mathieu equation. This can be seen from a detailed analysis of the Mathieu functions for arbitrary, but real, parameters (.alpha., q) . It happens that zones exist in the (.alpha., q)-plane in which .nu. is real and negative, so stable solutions exist. Other zones exist in which .nu. is complex, and at least one unstable solution exists. The first two stable zones are separated by a zone of instability connected at .alpha.=1, as shown in FIG. 2.
For .alpha.=1 and q &gt;0, which is the case of interest here, the solution lies in the first unstable region. Of course, .gamma. can deviate slightly from 2, the effect of which could lead to solutions in the stable zones. A typical solution for a stable wave is illustrated in FIG. 3, in which the wave amplitude is variable, but does not increase with distance. In fact, the wave amplitude displays "beats", because the wavelengths of the wall and the ultrasonic wave are slightly, but not grossly different.
Evidently, sufficiently large deviations from .gamma.=2 place .nu. in the stable regions above and/or below the first zone of instability; the boundary of the unstable zone is not strictly symmetrical. Conversely, sufficiently small deviations from .gamma.=2 do not allow the boundary of the unstable zone to be reached; the perturbed solution remains unstable in this case, depending on the size of q. In particular, a straight-forward thermodynamic argument reveals that for adiabatic walls the wave intensity grows at the expense of fluid internal energy, or temperature. As the wave propagates and grows, the gas is cooled slightly and .gamma. decreases, causing the characteristic exponent to eventually move into the first stable zone of FIG. 2. This results in no further amplification, unless the fluid is a good conductor of heat, in which case the temperature gradient is equilibrated and growth may continue.